\section{Preliminary orbit determination}

\begin{frame}{\thesection.\ \insertsection}
\begin{columns}
\column{0.6\textwidth}
    Now that we know how to describe the motion of an object in orbit, a significant question is how do we actually determine it.
\column{0.3\textwidth}
    \vspace{-20pt}
    \hspace{-20pt}\includegraphics[scale=0.6]{fig_2_8.pdf}
\end{columns}
\begin{itemize}
\item There are very sophisticated methods of orbit determination,
    based upon statistical approaches, in fact, entire books have been written about the subject.
\item However, all of these sophisticated methods are based upon correcting
    \textcolor{blue}{preliminary orbit estimates}.
\item Therefore, methods of initially determining the orbit of an object are relevant.
    \begin{itemize}
    \item[\mysquare] There are several methods (and variations upon these methods) to accomplish this,
       and they depend upon \textcolor{blue}{the measurements} that are available.
    \item[\mysquare] We shall examine three methods.
    \end{itemize}
\end{itemize}
\end{frame}

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\subsection{Orbit determination from three position vectors}
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\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Given three position vectors \(\vec{_{}r}_1, \vec{_{}r}_2, \vec{_{}r}_3\) and a time \(t_i\) for some $i=1,2,3$:
\begin{center}\includegraphics{fig_3_1.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.1:} Orbit determination from three position vectors\end{center}
\begin{block}{To obtain the orbital elements:}
1. Compute $r_1=\sqrt{\vec r_1 \cdot \vec r_1},r_2=\sqrt{\vec r_2 \cdot \vec r_2},r_3=\sqrt{\vec r_3 \cdot \vec r_3}$. \\
2. Compute \(\vec{n} = \vec{_{}r}_1 \times \vec{_{}r}_3\) and $n=\sqrt{\vec n \cdot \vec n}$.
\begin{center}\includegraphics{fig_3_p25.pdf}\end{center}
\end{block}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\begin{block}{}
3. Compute \(\alpha\) and \(\beta\)
\[
\alpha = \frac{\vec{n} \cdot (\vec{_{}r}_2 \times \vec{_{}r}_3)}{n^2}, \quad 
\beta = \frac{\vec{n} \cdot (\vec{_{}r}_1 \times \vec{_{}r}_3)}{n^2}
\]
4. Compute angular momentum vector
\[ h = \sqrt{\frac{\mu(r_2 - \alpha r_1 - \beta r_3)}{1 - \alpha - \beta}}, \quad 
\vec{h} = h\frac{\vec{n}}{n} \]
5. Compute eccentricity vector
\[
\vec{e} = \frac{1}{n^2} \left[ \left( \frac{h^2}{\mu} - r_1 \right) \vec{_{}r}_3 \times \vec{n} - \left( \frac{h^2}{\mu} - r_3 \right) \vec{_{}r}_1 \times \vec{n} \right]
\]
\end{block}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\begin{block}{}
6. Compute the velocity $\vec v_i$ corresponding to time \(t_i\):
\[ \vec{v}_i = \frac{\mu}{h^2} \vec{h} \times \left( \vec{e} + \frac{\vec{_{}r}_i}{r_i} \right) \]
7. Using \(\vec{_{}r}_i\), \(\vec{v}_i\) and \(t_i\), compute the orbital elements.
\end{block} 
\end{frame}

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\subsection{Orbit determination from three line-of-sight vectors}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
Suppose we only have optical measurement available.
\begin{itemize}
    \item In this case, we can only obtain the direction of the object (whose orbit we are trying to determine) from our observation location, but not the range (distance).
    \item Therefore, we cannot compute the position vector of the object.
    \item We must determine the orbit on the basis of line-of-sight vectors only.
\end{itemize}
\begin{center}\includegraphics{fig_3_2.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.2:} Orbit determination from three line-of-sight vectors\end{center}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\begin{center}\includegraphics{fig_3_2.pdf}\end{center}
\begin{center}\textcolor{blue}{Figure \arabic{section}.3:} Orbit determination from three line-of-sight vectors\end{center}
Let
\begin{description}
\item[$t_i$:] times, \(i = 1, 2, 3\), where \(t_1 < t_2 < t_3\).\\
\item[$\vec{_{}R}_i$:] the (known) position of the observer at time \(t_i\), \(i = 1, 2, 3\).\\
\item[$\vec{_{}l}_i$:] the (measured) unit line-of-sight vector at time \(t_i\), \(i = 1, 2, 3\).\\
\item[$\rho_i$:] the (unknown) range (distance to the object) at time \(t_i\), \(i = 1, 2, 3\).\\
\item[$\vec{_{}r}_i$:] the object orbital position at time \(t_i\), \(i = 1, 2, 3\).
\end{description}
\end{frame}

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\begin{block}{Assumption:}
\begin{enumerate}
    \item The true anomalies \(\theta_1\) and \(\theta_3\) correspond to \(\vec{_{}r}_1\) and \(\vec{_{}r}_3\) satisfying
    \vspace{-10pt}
    \[ \theta_3 - \theta_1 < 90^\circ \]
    \vspace{-24pt}
    \item The orbit is elliptical.
\end{enumerate}
\end{block}
\begin{block}{The known:}
\begin{enumerate}
    \item We are given line-of-sight measurements \(\vec{l}_1, \vec{l}_2, \vec{l}_3\) at times \(t_1\), \(t_2\) and \(t_3\).
    \item The position of observation is known at each of these times, and is given by $\vec{_{}R}_1,\vec{_{}R}_2$ and $\vec{_{}R}_3$.
\end{enumerate}
\end{block}
For the algorithm to obtain the orbital elements, please refer to:\\
A. H. J. de Ruiter, C. J. Damaren, J. R. Forbes, \textit{Spacecraft Dynamics and Control, an Introduction}, John Wiley and Sons Ltd, 2013, Page 108.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Orbit determination from two position vectors and time}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
    \begin{center}\includegraphics{fig_3_4.pdf}\end{center}
    \begin{center}\textcolor{blue}{Figure \arabic{section}.4:} Orbit determination from two position vectors and time\end{center}
\begin{block}{The known:}
\begin{enumerate}
    \item We are given positions \(\vec{_{}r}_1\) and \(\vec{_{}r}_2\), and the transfer time \(t_2 - t_1\).
\end{enumerate}
\end{block}
\begin{block}{Assumption:}
\begin{enumerate}
    \item The true anomalies \(\theta_2 - \theta_1 < 90^\circ\).
    \item The orbit is elliptical.
\end{enumerate}
\end{block}
For the algorithm to obtain the orbital elements, please refer to:\\
A. H. J. de Ruiter, C. J. Damaren, J. R. Forbes, \textit{Spacecraft Dynamics and Control, an Introduction}, John Wiley and Sons Ltd, 2013. Page 112.
\end{frame}

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\subsection{remarks}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{\thesection.\ \insertsection \\ \small\thesection.\thesubsection\ \insertsubsection}
\begin{itemize} \setlength{\itemsep}{20pt}
    \item The techniques presented here may be used to initially determine an orbit given some measurements.
    \item Other techniques may then be applied to improve the estimated orbit once additional measurements are available.
\end{itemize}
\end{frame}
